Templex : the tool to overstep the barrier of the fourth dimension

Christophe LETELLIER
Denise Sciamarella, Gisela Charo & Christophe Letellier

Attractors in three-dimensional state space can be described using template through knot theory : unfortunately, knot are all trivial in a 4-dimensional space and can no longer be used to discriminate the different type of chaotic attractors. Here, the strategy starts with the construction of a cell complex from a cloud of points in state space and then uses the flow to determine the sequence with which the cells are visited. The topology is described by a directed graph where the nodes are the cells of the complex. The procedure is nearly automatic and exhibit the ingredients involved in the structure of the trajectory within the state space. A box of tools as generatex, stripex, splitting line, joining line have been introduced - without any limitation related to dimensionality - to construct a templex - the tandem between the cell complex and the digraph - which encode the topology of the attractor investigated. Templexes open a systematic way to accurately classify chaotic attractors without any limitation regarding their dimension. The templex for a four-dimensional system initially investigated by Sciamarella [1] is shown in Fig. 1.

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Fig. 1. A templex for a four-dimensional system.

A notebook working with Mathematica for computing templex properties (homologies, orientability chain, generatex, stripex) from a given generating 2-complex and its associated digraph according to definitions proposed by Charo et al. [2]

[1] D. Sciamarella & G. B. Mindlin, Unveiling the topological structure of chaotic flows from data, Physical Review E, 64 (3), 036209, 2001.

[2] G. D. Charó, C. Letellier & D. Sciamarella, Templex : A bridge between homologies and templates for chaotic attractors, Chaos, 32 (8), 083108, 2022. Online

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